complexOfReal_deriv — Mathlib

English

A complex extension constructed from a real-linear map preserves differentiability properties in the complex setting.

Русский

Комплексное продолжение сохраняет дифференцируемость в комплексном контексте.

LaTeX

$$HasDerivAt f ( (fderiv Real f x) .complexOfReal h₂ ) x$$

Lean4

/-- In cases where the **Cauchy-Riemann Equation** guarantees complex differentiability at `x`, the
complex derivative equals the real derivative.
-/
theorem complexOfReal_deriv (h₁ : DifferentiableAt ℝ f x) (h₂ : fderiv ℝ f x I = I • fderiv ℝ f x 1) :
    deriv f x = fderiv ℝ f x 1 :=
  HasDerivAt.deriv (complexOfReal_hasDerivAt h₁ h₂)

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